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Stochastic processes on reductive homogeneous and symmetric spaces

  • Sonja Cox
  • Erwin Luesink
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Abstract

In this paper we present a method to effectively and efficiently provide pathwise approximations of the Riemannian Brownian motion on a reductive homogeneous space M≅ G/H endowed with a G-invariant metric. This setting includes spheres, hyperbolic spaces, the space of symmetric positive-definite matrices, flag manifolds and Stiefel manifolds. The simulation is based on Wong--Zakai approximations of an appropriately chosen SDE in G that is projected down onto M≅ G/H. The effectiveness lies in the fact that well-established results on the convergence rate of the Wong--Zakai equation immediately carry over to the homogeneous space thanks to Nash's embedding. The efficiency lies in the fact that explicit representations of the Wong--Zakai approximations can be obtained in this setting using the Lie exponential map.

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Cite

BibTeXRIS
BibTeX
@misc{cox2026stochastic,
  title = {{Stochastic processes on reductive homogeneous and symmetric spaces}},
  author = {Sonja Cox and Erwin Luesink},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.42025},
  eprint = {2610.42025},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.42025},
}
RIS
TY  - UNPB
TI  - Stochastic processes on reductive homogeneous and symmetric spaces
AU  - Sonja Cox
AU  - Erwin Luesink
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.42025
UR  - https://arxiv.org/abs/2610.42025
AB  - In this paper we present a method to effectively and efficiently provide pathwise approximations of the Riemannian Brownian motion on a reductive homogeneous space M≅ G/H endowed with a G-invariant metric. This setting includes spheres, hyperbolic spaces, the space of symmetric positive-definite matrices, flag manifolds and Stiefel manifolds. The simulation is based on Wong--Zakai approximations of an appropriately chosen SDE in G that is projected down onto M≅ G/H. The effectiveness lies in the fact that well-established results on the convergence rate of the Wong--Zakai equation immediately carry over to the homogeneous space thanks to Nash's embedding. The efficiency lies in the fact that explicit representations of the Wong--Zakai approximations can be obtained in this setting using the Lie exponential map.
ER  -